Thursday, July 17, 2008

Cats are the best....

I usually don't see dreams, and I see dreams about mathematics only very rarely. However, this night I had a dream about mathematics, and my cat managed to turn it into the most bizarre dream I ever had... The first part of this post is the story itself, hope you will find it funny, and the second part is about power series.

In my dream I had a conversation about sums of infinite power series. I have no idea why I would see a dream about them, it is not like I was thinking much about this. Just when the conversation turned to a specific simple problem, my cat decieded that he wants my pillow. In other words he came and sat down on it (I guess I should be thankful that he didn't sit on my head...). Naturally, this waked my up and I attempted to remove him, while still half emerged in the dream. This effectively turned the conversation I had in my dream in a conversation with my cat. Eventually I manged to convince him to get off (not before he tried to bite me for this).
I have always claimed that while dogs are "mens best friend", cats are "mathematician best friend". Now I am sure in this. Somewhere deep inside them hides a mathematical mind... :)

Photo by Per Ola Wiberg..(PO...or Powi)

Now, to the problem I mentioned. It shall be named the dreaming cat problem (for no reason whatsoever). The problem is as follows: Suppose you have an infinite sum:

Suppose also that the sum converges for all x such that |x|<>R. R is a fixed positive number. If you will write instead of x, M-x (M is a fixed number) what will happen, if anything? The answer - absolutely nothing interesting. The only thing that will change is that now the sum will converge for all x such that |M-x| to ? This is more interesting, we will now get that the series converges for |x|<=<.

This is result is not obvious, so it is time to proof all what I written. Firstly, lets proof that there is indeed such R. To do this we will use the root exam for infinite sums. This exam says that any given infinite sum converges if
(The limit is when n approaches infinity) and it doesn't converge if the limit is greater than one. If we will take this upper limit from the series we have we will get .

If we will suppose that the right side is equal to we will get that the sum converges for all |x|R.

To proof the second part, we will first need to proof a different statement. Given two different infinite sums and which converge for and , for what x the sum ) converges? Lets use the above method:

So we get that .
We can now use this formula to solve the problem. Lets suppose that for all n. Then . From this we will get that R in the above formula is 1. Now if we will divide by we will get that But is exactly R from the statement I wanted to proof, so we get that indeed this new series will converge for all |x|<=<.

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